Irregular conformal blocks and connection formulae for Painlev\'e V functions

Abstract

We prove a Fredholm determinant and short-distance series representation of the Painlev\'e V tau function $\tau(t)$ associated to generic monodromy data. Using a relation of $\tau(t)$ to two different types of irregular $c=1$ Virasoro conformal blocks and the confluence from Painlev\'e VI equation, connection formulas between the parameters of asymptotic expansions at $0$ and $i\infty$ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as $t\to 0,+\infty,i\infty$ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.